I'm having a little trouble figuring out how to start this. The question is as follows. At time $$t<0$$ a hydrogen atom is in the rest frame $$\mathfrak{R}$$. Then at time $$t=0$$ the atom suddenly starts moving at a speed $$V$$ which is non-relativistic. It is now described in the rest frame $$\mathfrak{R}'$$ under the transformation $$t' = t, \, \, \, \vec{r}' = \vec{r}-\vec{V}t$$ The question asks to find the probability that at time $$t>0$$ in rest frame $$\mathfrak{R}'$$ that the electron in the hydrogen atom is still in it's ground state.
I know so far that this is a time dependent perturbation theory. I also know the probability can be computed by using $$|C_m^{(1)}(t)|^2$$ where $$C_m^{(1)}(t) = -\frac{i}{\hbar}\int_0^t dt e^{i\omega_{m,i}t}H'_{m,i}(t)$$ and $$\omega_{m,i}= \frac{E_m-E_i}{\hbar}, \, \, \, H'_{m,i} = \langle\psi_m^0|\hat{H}'(t)|\psi_i^0\rangle$$ What I'm not clear on is what $$E_m$$, $$\psi_m^0$$ and $$\hat{H}'(t)$$ should be. Would these just be the Galilean transformations of $$E_i$$, $$\psi_i^0$$ and $$\hat{H}(t)$$ for a hydrogen atom, or is there something more subtle I'm missing here?